1. Introduction

Laser remote sensing involves the measurement of laser-beam transmission through the atmosphere and is subject to uncertainties caused by strong fluctuations resulting from physical mechanisms such as speckle, refractive turbulence, aerosol variability, detection noise, and background noise. Various applications of coherent lidar depend on estimating mean return power and these fluctuations frequently establish the limit of sensitivity by being the major cause of uncertainty in the average value of the lidar signal. The measurement uncertainty is defined as the normalized standard deviation of the lidar signals. In this study we consider the combined effects of atmospheric turbulence and transmitted-beam characteristics on the fluctuations of a signal that arises from the backscatter of a distant atmospheric volume element containing aerosol particles. Any others mechanisms producing lidar signal fluctuations, such as the prevalent speckle effect, are not the focus of this analysis and are not discussed here. All these very relevant considerations have been thought over carefully elsewhere. Our results regarding turbulence effects complement those earlier analyses.

The obtained results are used to gain a better understanding of the observed return signals using a coherent lidar system. Also, some measurements of potential importance are differential in nature (depolarization effects, differential absorption, target calibration), entailing comparison of the return power in two channels. In all cases, it is the ratio of received power in two channels that is to be averaged, so independent mean power measurements need to be supplemented by knowledge of the power fluctuation statistics.

Simulations of beam propagation in three-dimensional random media were used to study the effects of atmospheric refractive turbulence on coherent lidar performance. The approach provided the tools for analyzing laser radar with general refractive turbulence conditions, beam truncation at the telescope aperture, beam-angle misalignment, and arbitrary transmitter and receiver configurations. The simulation permitted the characterization of the heterodyne optical power [1] as well as the effect on lidar performance of the analytically intractable return variance from turbulent fluctuations [2]. In a first round of simulated measurements, we considered the lidar horizontally directed so it was possible to assume uniform turbulence with range. Most of the time, however, heterodyne lidars are aimed to profile the atmosphere parameters at altitudes well above the atmospheric layer adjacent to the underlying surface. In this study our simulation technique is extended to consideration of those non-uniform turbulence conditions presented when a ground lidar is profiling the atmosphere along vertical paths and slant paths with large elevation angles. As we will argue later, ground systems working under non-uniform conditions present characteristics markedly different to those found in our previous studies.

2. Heterodyne optical power for non-uniform turbulence conditions

The scheme of wavefront matching for optical heterodyning in coherent lidar systems can be outlined in simple terms. In the single-mode detection regime the scattered radiation field U_S(w, -f, R) from an atmospheric layer at range R and the reference local oscillator field U_LO(w, -f) are combined and focused onto the photodetector sensitive element placed on the focal plane w of one lens with focal on the propagation axis point -f The equation for the optical heterodyne power P expresses the performance of the heterodyne lidar in terms of the degree of coherence of the backscattered radiation and its proper match with the field of the local oscillator. Refractive turbulence effects are considered in the mutual coherence function of the backscattered field. However, to overcome the complexity of calculating the propagation of the random fields to the receiver, one embeds double-pass effects associated with lidar into the overlap of the transmitted and back-propagated local oscillator (BPLO) irradiances at the target plane p [3]:

where λ is the optical wavelength of the transmitted laser. For convenience, we express the lidar optical power P(R) by using free-space irradiances j_T and j_BPLO normalized to the laser <P_L> and the local-oscillator <P_LO> average power, respectively. Along with these normalization parameters, the term C(R) groups the conversion efficiencies that describe the various system component and the atmospheric scattering conditions (extinction and backscattering). As we are mainly concerned with the effects of the refractive turbulence, those parameters are mostly irrelevant here [2].

This formulation, which encompasses the transmitter and BPLO beams in the target plane, is physically explained on application of the reciprocity theorem [4, 5]. Especially relevant to this analysis, in order to extent the reciprocity theorem to propagation in atmospheric turbulence it is not necessary to ask for uniform atmosphere and, consequently, applies to any of the vertical and slant paths considered in this study.

Now, the equation for the optical heterodyne power P is not expressed anymore in terms of the degree of coherence of the backscattered radiation and its proper match with the field of the local oscillator onto the photomixer plane w, but in the overlap integral of the transmitted laser beam an virtual back-propagated local oscillator in the target plane p. The integral of Eq. (1) has to be evaluated over the target area. With this formulation, the problem of heterodyne lidar performance in the presence of atmospheric turbulence is reduced to one of computing intensity along the propagation paths.

In general, for any coherent lidar application relying on power measurement, an accurate knowledge of the statistics of the coherent power turbulent fluctuations as a function of range R is required if we are bound to take on the estimation of relative error in the measurement. From Eq.(1), it results apparent that the statistical properties of the signal P are those corresponding to the overlap integral of the transmitted and virtual back-propagated local oscillator [2]. In most cases, the error analysis would need to estimate the normalized power variances or, more generally, the normalized covariances. Involving high-order statistical moments of the field, no simple analytical solutions to the statistics of the heterodyne power have been described that we can use to describe our problem. Although theoretical calculations of beam propagation and the higher moments of the field are still difficult, some partial results on heterodyne statistics have been obtained for simplified beam configurations and idealistic atmospheric characterization [6]. However, the overlap integral of the transmitted and virtual back-propagated local oscillator can be easily statistically characterized by using simulation techniques.

The standard simulation technique is based on the Fresnel approximation to the wave propagation and models the atmosphere as a set of two-dimensional Gaussian random phase screens [7–9]. The treatment of turbulence is built on the generation of random phase screens at each calculation step along the propagation path. In our time-independent calculations, we need to use just space correlated phase screens to properly describe the statistical moments of the power received by our coherent system. At each given position they are generated anew for each successive simulation run. Phase screens are generated via a standard number generator used to compute pseudorandom arrays of statistically independent Gaussian numbers containing zero spatial correlation. By using the turbulence spectrum to filter the Fourier transform of the uncorrelated arrays and inverse transforming the result, one obtains two-dimensional spatial fields of phase fluctuations with the right spatial correlation.

All simulations will assume non-uniform turbulence with range, i.e., slant propagation paths, and use the Hill turbulence spectrum [10] -with typical inner scale l₀ of 1 cm and realistic outer scale L₀ of the order of 5 m- to describe the spatial correlation of the phase screens. Although similar conclusions on the lidar basic behavior could be obtained by using simpler spectrum models, the Hill spectrum is widely considered to be more accurate representing the high frequency features of the refractive-index fluctuations. In order to accommodate the characteristics of the turbulence spectrum, the simulation technique uses a numerical grid of 1024×1024 points with 5-mm resolution [9].

The turbulence strength is included in the atmospheric spectrum by means of the well-known structure parameter $C_n^2$. As it was commented before, for applications involving propagation along a horizontal path, it is customary to assume that the structure parameter $C_n^2$ remains essentially constant. Propagation along a vertical or slant path, however, requires a $C_n^2$ (h) profile model to describe properly the varying strength of optical turbulence as a function of altitude h. Although several different turbulence profile models are up for consideration, in this study we have used the well-known Hufnagel-Valley (H-V) model to describe all the considered atmospheric profiles [11]. This model allows accommodating with ease any local near-ground turbulence conditions $C_{n\mathit{0}}^{\mathit{2}}$ and any rms high-altitude wind speed v (pseudowind), the two basic parameters needed to model the dependency of turbulence along a slant path. In general, the ground turbulence level has little effect above 1-km altitude and the wind speed governs the profile behavior primarily in the vicinity of 10 km. Although of importance in making temporal calculations, the wind speed is not relevant to our time-independent estimations and we will use, without loss of generality, the H-V profile model with a standard v=21 m/s in all the situations considered in this study. Also, as mainly concerned with the influence of turbulence in ground-based lidar systems, we will use realistically several different turbulence levels near the ground. For applications involving satellite-borne lidar instruments, the influence of high-altitude wind speeds may result increasingly important.

To model our continuous, extended, non-uniform medium with multiple phase screens, we need to assume that no intensity fluctuations are produced over the interscreen distance ΔR and that we can model the effect of the medium as purely an addition of phase [7]. As the number of screens is finite, this is not strictly true, but we may impose the condition that the scattering be weak over the interscreen distance ΔR, i.e. ${\mathrm{\sigma }}_{\mathrm{I}}^2$(ΔR)<0.1. Here, ${\mathrm{\sigma }}_{\mathrm{I}}^2$ is defined as the beam intensity variance calculated by using the Rytov theory [12]. Although the Rytov theory produces an accurate estimate of the normalized intensity variance only when turbulence is weak, ${\mathrm{\sigma }}_{\mathrm{I}}^2$ is still a useful parameter to specify the scattering strength relevant to a given simulated experiment [9]. If the simulation is to represent an extended medium even in weak scatter when ${\mathrm{\sigma }}_{\mathrm{I}}^2$ (R)<0.1, we must in addition require that the path R to be sampled often enough. The condition is met by requiring that less than 10% of the total scintillation be allowed to take place over the interscreen distance, i.e. ${\mathrm{\sigma }}_{\mathrm{I}}^2$ (ΔR) < 0.1${\mathrm{\sigma }}_{\mathrm{I}}^2$ (R). Thus, for a given value of scintillation over the whole path, these conditions determine a set of non-equidistant screens for any possible turbulence profile. For the utmost atmospheric altitude considered in our modeling, the technique simulates a continuous, non-uniform random medium with a minimum of 30 (for the shortest, vertical path) and as much as 50 (for slant paths with low elevation angles) two-dimensional phase screens.

Some previous works dealing with the problem of the numerical analysis of the effect of refractive turbulence statistics in the atmosphere [13] have used an algorithm based on the simulation of only one phase screen. In this approach, the atmosphere is modelled as a single atmospheric layer adjacent to the earth surface with an effective thickness ΔR_eff of several hundred meters. Unfortunately, this approach is just valid when the sounding range R is much larger than the layer thickness (R>>ΔR_eff). That limits the applicability of the tactic to ranges well above the underlying ground. In any other practical situation, as described in the previous paragraphs, where we may be interested in the lidar performance not only at ranges very far from the ground but also in its proximity, we need to sample the atmospheric turbulence vertical profile with a much higher level of detail.

Errors produced by statistical fluctuation of the measured field moments are reduced by simply running an increased number of realizations. In any of the scenarios considered in this study, we run over 4000 samples to reduce the statistical uncertainties of our estimations to less than 2% of their corresponding mean values.

3. Uncertainty on coherent measurements along slant paths

Heterodyne lidar simulated measurements were taken with the lidar elevated at angles θ ranging from 0° to 90°. As a first round of horizontal simulated measurements were already presented previously elsewhere [2, 14], in this study we analyze vertical (90° elevation angle) and slant paths (30° and 60° elevation angles). For profiling purposes, vertical pointing provides the best maximum range capability; however, height resolution (equal to range resolution times the sine of the elevation angle) is limited by the transmit pulse length. It is usual to lower elevation angles to increase the vertical resolution of the measurements.

Figures 1–2 show the statistics of the coherent power fluctuations as a function of altitude h of a realistic monostatic lidar system. Both the mean coherent power -normalized such that at the shortest range is 0 dB- and the standard deviation of coherent power fluctuations are pictured in the figures for different moderate-to-strong refractive turbulence ${\mathrm{C}}_{\mathrm{n}}^2$ daytime levels. We use Eq. (1) to compute our estimations. Two wavelengths, 2 μm (Fig .1) and 10 μm (Fig. 2), and several propagation paths have been considered in the figures. Transmitted and virtual LO beams were assumed to be matched, collimated, perfectly aligned, Gaussian, and truncated at a telescope aperture of typical diameter D=16 cm. The beam truncation was 1.25 (i.e., D = 1.25 × 2ω₀ , where ω₀ is the 1/e² beam irradiance radius). This truncation maximizes the coherent system efficiency in the ideal case of absence of turbulence [15]. In any situation regarded in this study, the coherent power standard deviation results are generally below 0.5 (i.e., less than -3-dB power fluctuation around the mean values).

The structural characteristic of refractive index in the atmospheric boundary layer essentially decreases when the height h increases. Therefore, the main distortions of a laser beam that propagate along a vertical (h=R) or a slant (h=R sinθ) path occur in the atmospheric layer adjacent to the underlying surface. When average heterodyne optical power is considered in Fig. 1 (left column), we can appreciate how the effects of refractive turbulence enhancement are important just for altitudes smaller than 1.5 km. This already well-known enhancement is physically caused by turbulence-induced irradiance fluctuations at the target plane [1]: Turbulence causes small bright spots on the scattering target, increasing the value of the overlap integral in Eq. In the same way, for upper altitudes, the predominant turbulent mechanism becomes beam spreading, which steadily reduces the performance. The beam spread decreases the overall lidar coherent optical power under any turbulence conditions.

When uncertainty on coherent measurements along slant paths is considered in Fig. 1 (right column), it is apparent that a maximum of the variance occurs at low altitudes, where the partial focusing of the beam on the target plane produces the turbulence enhancement. Roughly, at this altitudes, just a very few scintillation spots fill the beam area on the scattering target and, consequently, the so-called averaging principle is almost negligible [1,2]. For larger ranges, the beam resolves several scintillations, producing an intense averaging effect: The irradiance-related integral in Eq. (1) is evaluated in the target plane over the beam area, so a large number of bright spot scintillations, caused by either the small area of each spot or a large beam area, tends to reduce power fluctuations. For strong-turbulence conditions and large propagation paths, the size of the laser beam at the target no longer depends on the transmitter aperture. The spot area, and thus the beam-averaging effect, is defined by the turbulence beam spreading. Also, at the same ranges the beam intensity fluctuations saturate, establishing a limit to the continuously increasing scintillations that occurs in the weak turbulence regime. These multiple effects sum up to produce the decreasing of power standard deviation with altitude observed in the figures. Notably, at higher altitudes power standard deviation tends to converge towards values close to 0.4 with independency of elevation angles and turbulence levels considered in the analysis. It is interesting to remark, however, that, when strong turbulence (C_n0²=10^-12 m^-2/3) and small elevation angles (30°) are considered in Fig. 1, after a short decrease the standard deviation of the turbulence fluctuations increases again with altitude. The reason here is likely to be that beam spreading becomes comparatively less important than scintillation. In this regime, the saturation effect and the averaging over the reduced beam size are no longer able to compensate for the trend of the fluctuations to increase with the range.

 

Fig. 1. Statistics of the coherent power turbulent fluctuations as a function of altitude h for a 2-μm wavelength, 16-cm aperture, monostatic lidar system. The Hufnagel-Valley C_n²(h) profile model with different moderate-to-strong near-ground refractive turbulence C_n0² conditions is considered. Both the mean coherent power (left column) and the coherent power standard deviation (right column) are shown for several vertical (90° elevation angle) and slant (30° and 60° elevation angles) propagation paths.

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Fig. 2. Similar to Fig.1 but for a 10-μm monostatic lidar. Again, both the mean coherent power (left column) and the coherent power standard deviation (right column) are shown for several vertical and slant paths.

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Interestingly, in a most unfavorable situation than those considered in the figures, with ground lidar systems profiling the atmosphere along horizontal paths, the accumulated turbulence level and its effects was shown to be just slightly larger. Once again, beam averaging may help to elucidate these results. Although accumulated turbulence for horizontal paths is much higher than for any other possible propagation situation, and consequently we may expect an intense power uncertainty, much of the effect is effectively wipe out by the averaging associated to large turbulence beam spreading. Vertical and slant paths don’t undergo turbulence effects as severe as those observed in horizontal paths, but neither they can make an effective use of a large averaging.

The performance of the 10-μm lidar is shown in Fig. 2, in a format similar to that of Fig. 1. Compared with those of the 2-μm lidar, the turbulence effects are apparently less, but the same physical features can be observed. Enhancement appears at shorter ranges and, for longer ranges, the predominant turbulent mechanism becomes beam spreading, which, once again, steadily reduces the lidar performance. As stated in the figures, standard deviation power fluctuations are larger for the 2-μm system (Fig. 1), where we may expect more sensitivity to turbulence, than for the 10-μm lidar (Fig. 2). When horizontal paths were considered [2], surprisingly we observed the opposite circumstances, i.e. power fluctuations larger for the 10-μm case than for the 2-μm situation. We explained this result of our simulations by considering the fact that the beam averaging for the 10-μm system was remarkably smaller as a consequence of larger beam intensity scales in the target plane reducing the ability for leveling signal fluctuations. Now, however, when slant propagation paths are taken into consideration, turbulence effects on 2-μm lidars are not already intense enough to produce averaging phenomena able to wipe out an important part of the power fluctuations. In any case, when 10-μm and strong turbulence is considered (Fig.2, upper row), power standard deviations are still comparable to those observed for the 2-μm occurrence and similar turbulence levels (Fig.1, upper row).

4. Final remarks

The inherent statistic uncertainty of coherent return fluctuations have been estimated for ground lidar systems profiling the atmosphere along slant paths with large elevation angles. Although less intense, fluctuations in received power owing to turbulence produce outcomes in the performance of any coherent system similar to those that result from speckle and, consequently, a precise description of this effect is needed to fully characterize the performance of heterodyne lidars in the atmosphere. The analytically intractable problem of describing the coherent return variance was considered by simulation of beam propagation in a realistic way. One advantage of the beam simulation is that the structure of the beam can be examined at any point, which in this study allows the explanations to the observed phenomena to be verified readily.

The results for the coherent power standard deviation indicate the presence of a maximum at shorter ranges when strong turbulence levels are considered. As expected, in any case the magnitude of the normalized standard deviation is not as large as that resulting from horizontal paths but, surprisingly, the differences are not very apparent. Beam averaging is a simple way of interpreting these results. Still, although we observe that these turbulence-induced fluctuations are not able to provoke power fading as large as that associated with speckle (0-dB power fluctuations around the mean values), for most turbulence levels it is significant at any of the elevation angles considered in this study.

All the considerations of this work will help us to understanding the most comprehensive problem of coherent lidar uncertainties on measurements, such as differential absorption, which are differential in nature and are becoming critically important in many atmospheric studies.

This research was partially supported by the Spanish Department of Science and Technology MCYT grant No. REN 2003-09753-C02-02.